Term Rewriting System R:
[x]
f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(s(x))) -> F(f(x))
F(s(s(x))) -> F(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

F(s(s(x))) -> F(x)
F(s(s(x))) -> F(f(x))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(x))) -> F(f(x))
two new Dependency Pairs are created:

F(s(s(x''))) -> F(s(x''))
F(s(s(s(s(x''))))) -> F(s(f(f(x''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(x''))))) -> F(s(f(f(x''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(x''))))) -> F(s(f(f(x''))))
three new Dependency Pairs are created:

F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(s(s(x'''))))) -> F(s(f(s(x'''))))
F(s(s(s(s(s(s(x'))))))) -> F(s(f(s(f(f(x'))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(x'))))))) -> F(s(f(s(f(f(x'))))))
F(s(s(s(s(x'''))))) -> F(s(f(s(x'''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x))) -> F(x)
F(s(s(x''))) -> F(s(x''))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(x'''))))) -> F(s(f(s(x'''))))
two new Dependency Pairs are created:

F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(x'))))))) -> F(s(f(s(f(f(x'))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(x'))))))) -> F(s(f(s(f(f(x'))))))
four new Dependency Pairs are created:

F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(f(s(s(f(x''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(f(s(f(s(x''))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(f(s(f(f(x''))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(f(s(f(s(x''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(f(s(s(f(x''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(x''))))))) -> F(s(f(s(s(f(x''))))))
four new Dependency Pairs are created:

F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
F(s(s(s(s(s(s(s(s(x'))))))))) -> F(s(f(s(s(s(f(f(x'))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(s(s(x'))))))))) -> F(s(f(s(s(s(f(f(x'))))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(f(s(f(s(x''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(f(s(f(f(x''))))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(x''))))))) -> F(s(f(s(f(s(x''))))))
three new Dependency Pairs are created:

F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
F(s(s(s(s(s(s(s(x')))))))) -> F(s(f(s(s(f(f(x')))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 7
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(s(x')))))))) -> F(s(f(s(s(f(f(x')))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(s(s(x'))))))))) -> F(s(f(s(s(s(f(f(x'))))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(f(s(f(f(x''))))))))
five new Dependency Pairs are created:

F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(s(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(s(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(f(s(x'''))))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(f(s(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(s(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(s(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(s(s(x'))))))))) -> F(s(f(s(s(s(f(f(x'))))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(s(x')))))))) -> F(s(f(s(s(f(f(x')))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
two new Dependency Pairs are created:

F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 9
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(f(s(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(s(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(s(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(x')))))))) -> F(s(f(s(s(f(f(x')))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(s(s(x'))))))))) -> F(s(f(s(s(s(f(f(x'))))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(s(s(x'))))))))) -> F(s(f(s(s(s(f(f(x'))))))))
five new Dependency Pairs are created:

F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(s(s(s(f(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(f(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(s(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(f(s(x''))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 10
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(f(s(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(s(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(f(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(s(s(s(f(f(x''))))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(f(s(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(s(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(s(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(x')))))))) -> F(s(f(s(s(f(f(x')))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(x'''))))))) -> F(s(f(s(s(s(x'''))))))
two new Dependency Pairs are created:

F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 11
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(f(s(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(s(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(f(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(s(s(s(f(f(x''))))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(f(s(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(s(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(s(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(x')))))))) -> F(s(f(s(s(f(f(x')))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(s(x')))))))) -> F(s(f(s(s(f(f(x')))))))
five new Dependency Pairs are created:

F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(s(s(f(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(f(f(f(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(s(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(f(s(x'')))))))
F(s(s(s(s(s(s(s(s(s(x'')))))))))) -> F(s(f(s(s(f(s(f(f(x'')))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 12
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(s(s(s(x'')))))))))) -> F(s(f(s(s(f(s(f(f(x'')))))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(f(s(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(s(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(f(f(f(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(s(s(f(f(x'')))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(f(s(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(s(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(f(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(s(s(s(f(f(x''))))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(f(s(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(s(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(s(s(f(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(s(s(f(f(x'''))))))))
five new Dependency Pairs are created:

F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(s(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(f(f(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(s(s(f(s(f(f(x'))))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 13
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(s(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(f(f(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(s(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(f(s(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(s(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(f(f(f(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(s(s(f(f(x'')))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(f(s(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(s(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(f(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(s(s(s(f(f(x''))))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(f(s(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(s(f(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(s(s(s(x'')))))))))) -> F(s(f(s(s(f(s(f(f(x'')))))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(s(f(x'''))))))))
five new Dependency Pairs are created:

F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(f(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(f(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(f(s(s(s(x''''))))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(s(s(f(f(x'))))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 14
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(s(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(f(s(s(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(f(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(f(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(f(f(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(s(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(s(x'')))))))))) -> F(s(f(s(s(f(s(f(f(x'')))))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(f(s(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(s(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(f(f(f(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(s(s(f(f(x'')))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(f(s(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(s(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(f(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(s(s(s(f(f(x''))))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(f(s(x'''))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(s(s(f(s(f(f(x'))))))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(f(s(f(s(f(s(x'''))))))))
four new Dependency Pairs are created:

F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(f(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(f(s(s(s(x''''))))))))
F(s(s(s(s(s(s(s(s(s(x')))))))))) -> F(s(f(s(f(s(s(f(f(x')))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 15
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(s(s(s(s(s(s(s(s(x')))))))))) -> F(s(f(s(f(s(s(f(f(x')))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(f(s(s(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(f(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(f(s(s(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(f(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(f(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(s(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(f(f(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(s(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(s(x'')))))))))) -> F(s(f(s(s(f(s(f(f(x'')))))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(f(s(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(s(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(f(f(f(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(s(s(f(f(x'')))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(f(s(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(s(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(f(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(s(s(s(f(f(x''))))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(s(s(f(f(x'))))))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(f(s(f(f(x'))))))))))
six new Dependency Pairs are created:

F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(s(s(f(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(f(s(s(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(f(s(f(s(s(f(x''))))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(f(s(f(s(f(s(x''))))))))))
F(s(s(s(s(s(s(s(s(s(s(s(s(x''))))))))))))) -> F(s(f(s(f(s(f(s(f(s(f(f(x''))))))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 16
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(s(s(s(s(s(s(s(s(s(s(s(s(x''))))))))))))) -> F(s(f(s(f(s(f(s(f(s(f(f(x''))))))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(f(s(f(s(f(s(x''))))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(f(s(f(s(s(f(x''))))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(f(s(s(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(s(s(f(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(f(s(s(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(f(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(f(s(s(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(f(s(s(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(f(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(f(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(s(s(x'))))))))))) -> F(s(f(s(s(s(f(s(f(f(x'))))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(f(s(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(f(s(s(s(s(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(f(f(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(x''''))))))))) -> F(s(s(s(s(s(f(f(x''''))))))))
F(s(s(s(s(s(s(s(s(s(x'')))))))))) -> F(s(f(s(s(f(s(f(f(x'')))))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(f(s(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(f(s(s(s(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(f(f(f(f(x'')))))))
F(s(s(s(s(s(s(s(x'')))))))) -> F(s(s(s(s(f(f(x'')))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(s(s(x''))))))))))) -> F(s(f(s(s(s(f(s(f(f(x''))))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(f(s(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(f(s(s(s(s(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(f(f(s(f(f(x''))))))))
F(s(s(s(s(s(s(s(s(x''))))))))) -> F(s(s(s(s(s(f(f(x''))))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(f(f(s(x''''))))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(s(x''''))))))
F(s(s(s(s(s(s(s(s(x'''))))))))) -> F(s(s(s(f(s(f(f(x'''))))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(f(s(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(f(f(f(x'''))))))
F(s(s(s(s(s(s(x'''))))))) -> F(s(s(s(s(f(x'''))))))
F(s(s(s(s(s(s(x''))))))) -> F(s(s(s(f(f(x''))))))
F(s(s(s(s(s(x')))))) -> F(s(s(f(f(x')))))
F(s(s(s(s(x''''))))) -> F(s(s(s(x''''))))
F(s(s(s(s(x'''))))) -> F(s(s(f(x'''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)
F(s(s(s(s(s(s(s(s(s(x')))))))))) -> F(s(f(s(f(s(s(f(f(x')))))))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))




Termination of R could not be shown.
Duration:
0:50 minutes